EP3631763B1 - Verfahren und vorrichtung zur bildrekonstruktion - Google Patents

Verfahren und vorrichtung zur bildrekonstruktion Download PDF

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EP3631763B1
EP3631763B1 EP18805847.3A EP18805847A EP3631763B1 EP 3631763 B1 EP3631763 B1 EP 3631763B1 EP 18805847 A EP18805847 A EP 18805847A EP 3631763 B1 EP3631763 B1 EP 3631763B1
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functional
image data
image
processing device
image processing
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EP3631763A1 (de
EP3631763A4 (de
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Mats Persson
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Prismatic Sensors AB
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T11/00—2D [Two Dimensional] image generation
    • G06T11/003—Reconstruction from projections, e.g. tomography
    • G06T11/008—Specific post-processing after tomographic reconstruction, e.g. voxelisation, metal artifact correction
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T11/00—2D [Two Dimensional] image generation
    • G06T11/003—Reconstruction from projections, e.g. tomography
    • G06T11/006—Inverse problem, transformation from projection-space into object-space, e.g. transform methods, back-projection, algebraic methods
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N23/00—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00
    • G01N23/02—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by transmitting the radiation through the material
    • G01N23/04—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by transmitting the radiation through the material and forming images of the material
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N23/00—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00
    • G01N23/02—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by transmitting the radiation through the material
    • G01N23/06—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by transmitting the radiation through the material and measuring the absorption
    • G01N23/083—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by transmitting the radiation through the material and measuring the absorption the radiation being X-rays
    • G01N23/087—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by transmitting the radiation through the material and measuring the absorption the radiation being X-rays using polyenergetic X-rays
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2211/00—Image generation
    • G06T2211/40—Computed tomography
    • G06T2211/408—Dual energy
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2211/00—Image generation
    • G06T2211/40—Computed tomography
    • G06T2211/416—Exact reconstruction
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2211/00—Image generation
    • G06T2211/40—Computed tomography
    • G06T2211/424—Iterative

Definitions

  • the present disclosure relates to methods, devices and computer programs for reconstructing image data based on performed x-ray measurements.
  • the photon counts may for example be obtained by direct readout pf a photon-counting x-ray detector.
  • the photon counts may be the output of a post processing scheme operating on the counts obtained by readout from a photon-counting detector.
  • the post-processing scheme may for example involve summation, filtering, averaging, and application of correction factors or correction terms.
  • the image data may consist of a single image, consisting of for example one of the basis coefficients or a combination thereof.
  • the image data may consists of coefficients in a non-pixelized representation of an image.
  • the elements of a may be Fourier coefficients, or wavelet coefficients, or coefficients in a representation of the image as a sum of blobs.
  • the vector a is, in a typical case, obtained through the optimization of a function ⁇ ( a , m ) of a and m .
  • a function which is optimized to find a will be referred to as a functional.
  • Such a functional which is a function of measured data in the projection domain, i.e. image data relating to the transmission through or photon count values after an object, will be referred to as a projection-based functional.
  • a projection-based functional may be used for image-based material decomposition or reconstruction.
  • a projection-based functional may be a function of line integrals of the linear attenuation coefficient or of basis coefficients along a projection line.
  • Such a functional can be used to perform image-based material decomposition or reconstruction.
  • a functional ⁇ (a, m) may incorporate prior information about the imaged object.
  • this prior information may be provided in the form of an edge-preserving regularizer, which penalizes rapid variations in the image.
  • the prior information may be provided in the form of a discrepancy term penalizing a difference between the reconstructed image and a prior image.
  • N 0 ( E ) is the incident spectrum
  • S i ( E ) is a weight function modeling the sensitivity of the energy bin to different incident energy levels
  • s ik is the number of expected scatter counts from the imaged object.
  • B may be represented by a sparse matrix with elements that are nonzero only for nearest-neighboring pixels.
  • the elements of B may decrease with increasing distance between pixels.
  • B may be obtained from a Monte Carlo simulation of a pencil beam of photons impinging on one detector element and scattered into neighboring detector elements.
  • Eq. (1) does not include pile-up.
  • ⁇ ik p may be given by a paralyzable model or a non-paralyzable detector model.
  • MAP maximum a posteriori
  • Modern CT reconstruction algorithms typically generate the reconstructed image as a maximum a posteriori (MAP) estimate of the image given the measured data.
  • the MAP estimate may build on a complete model of the relationship between the registered counts and the image values, or it may build on a simplified model of the relationship in order to simplify the optimization algorithm.
  • MAP reconstruction reduces noise and allows correction for detrimental effects such as scatter and optical blur.
  • ⁇ ik p includes the effect of cross-talk and pileup is obtained from (1) with the blur operator B and the pileup function f applied.
  • R ( a ) is an edge-preserving regularizer, which penalizes differences between neighboring detector elements.
  • the expression that is optimized in (2) will be referred to as an MAP functional.
  • (2) becomes a maximum likelihood (ML) estimator, and the functional to be minimized is called a maximum likelihood functional Since (2) is difficult to solve fast enough, it is common practice to replace it with a simplified penalized weighted least squares estimator.
  • T denotes the forward ray transform operator and ⁇ jk is an estimate of the line integral
  • a jk ⁇ l k a j d l along projection ray k.
  • ⁇ 2 ( ⁇ jk ) is the variance of ⁇ jk .
  • ⁇ jk can be obtained from the measured counts m ik for each individual detector element using maximum likelihood estimation or a look-up table.
  • Eq. (3) can be computed quickly using e.g. the iterative coordinate descent (ICD) method or the separable quadratic surrogates (SQS) method, but it gives inferior image quality compared to (2) since it builds on a simplified noise model and ignores detector cross-talk and object scatter.
  • ICD iterative coordinate descent
  • SQL separable quadratic surrogates
  • (3) is based on modelling the noise as Gaussian instead of Poisson, which is an approximation.
  • US patent 5390258 relates to a method of acquiring an image from an object, wherein a set of training images is used to generate a convergent series expansion, and wherein the measured signals are used to generate a truncated series expansion of an image of the object.
  • US patent 6754298 relates to a method based on a statistical model for reconstructing images from transmission measurements with high energy diversity.
  • US patent 7551708 relates to a method of reconstructing material decomposed images from data from energy discriminating computed tomography detectors using the iterative coordinate descent (ICD) algorithm.
  • ICD iterative coordinate descent
  • US patent 9165384 relates to a method of image reconstruction which reconstructs a plurality of final component images of an object based on spectral projection data, wherein intermediate images are used in the reconstruction algorithm and wherein correlations between these intermediate images are taken into account in the algorithm.
  • US patent 8923583 relates to a tomographic reconstruction method wherein material component images are reconstructed by optimizing a joint likelihood functional, which includes information on correlations between the component sinograms.
  • US patent application 2016120493A1 relates to an x-ray CT image processing method wherein a joint posterior distribution, based on a prior probability distribution, is used to estimate an x-ray absorption coefficient from measurements with different wavelengths.
  • US patent 8929508 relates to a method of computing line integrals of basis coefficients through an object from x-ray photon transmission measurements by computing a first approximation to the line integrals and combining the first approximation with a correction computed from a calibration phantom.
  • US patent 6907102 relates to a method of image iterative reconstruction wherein a cross-section reconstruction vector approximately matching the projection data is determined using a computed tomography model.
  • US patent 7885371 relates to a method of tomographic image reconstruction wherein a first reconstruction method, which converges faster on low spatial frequencies than on high spatial frequencies, is followed by a second reconstruction method which converges faster on high spatial frequencies than on low spatial frequencies.
  • US patent 9508163 relates to a method of iterative tomographic reconstruction wherein each iteration of an outer loop includes iterative processing of an inner loop.
  • US patent 9020230 relates to a reconstruction method employing an outer iteration loop and an inner iteration loop, wherein the inner iteration loop calculates a preconditioner used by the outer loop.
  • US patent 6256367 relates to a method of correcting for artifacts due to scatter CT images by Monte Carlo simulating photon scatter and subtracting the simulated photon energy from the measured projection data.
  • WO 2016/148616 relates to x-ray imaging and corresponding imaging reconstruction and imaging tasks using on energy-resolved image data from a photon-counting multi-bin detector or an intermediate storage.
  • WO 2016/148616 involves processing the energy-resolved image data by performing at least two separate basis decompositions using different number of basis functions for modeling linear attenuation.
  • a first basis decomposition is performed using a first smaller set of basis functions to obtain at least one first basis image representation, and wherein a second basis decomposition is performed using a second larger set of basis functions to obtain at least one second basis image representation.
  • the methodology further involves reconstructing a first image based on said at least one first basis image representation obtained from the first basis decomposition, and combining the first image with information representative of said at least one second basis image representation.
  • an image processing device for reconstructing image data from x-ray data measured with an imaging system comprising at least one photon-counting detector, according to claim 7.
  • a computer program adapted to reconstruct image data from x-ray data measured with an imaging system comprising at least one photon-counting detector, according to claim 14.
  • the proposed technology provides for a fast algorithm that yields high quality image data.
  • the provided algorithm has the virtue of being able to generate an image that takes several physical effects into account, such as non-Gaussian statistics, detector cross-talk, pile-up and optical blur.
  • the provided algorithm is also fast since it builds on solving a simplified optimization problem as a first step and then correcting the resulting first image with a few computationally inexpensive steps.
  • the x-ray imaging system 100 basically comprises an x-ray source 10 , an x-ray detector system 20 and an associated image processing device 30 .
  • the x-ray detector system 20 is configured for registering radiation from the x-ray source 10 that may have been focused by optional x-ray optics and passed an object or subject or part thereof.
  • the x-ray detector system 20 is connectable to the image processing device 30 via suitable analog processing and read-out electronics (which may be integrated in the x-ray detector system 20 ) to enable image processing and/or image reconstruction by the image processing device 30 .
  • an x-ray imaging system 100 comprises an x-ray source 10 , which emits x-rays; an x-ray detector system 20 , which detects the x-rays after they have passed through the object; analog processing circuitry 25 , which processes the raw electrical signal from the detector and digitizes it; digital processing circuitry 40 which may carry out further processing operations on the measured data such as applying corrections, storing it temporarily, or filtering; and a computer 50 which stores the processed data and may perform further post-processing and/or image reconstruction.
  • a challenge for x-ray imaging detectors is to extract maximum information from the detected x-rays to provide input to an image of an object or subject where the object or subject is depicted in terms of density, composition and structure. It is still common to use film-screen as detector but most commonly the detectors today provide a digital image.
  • Modern x-ray detectors normally need to convert the incident x-rays into electrons, this typically takes place through photo absorption or through Compton interaction and the resulting electrons are usually creating secondary visible light until its energy is lost and this light is in turn detected by a photo-sensitive material.
  • detectors which are based on semiconductors and in this case the electrons created by the x-ray are creating electric charge in terms of electron-hole pairs which are collected through an applied electric field.
  • the x-ray photons including also photons after Compton scattering, are converted to electron-hole pairs inside a semiconductor detector, where the number of electron-hole pairs is generally proportional to the photon energy.
  • the electrons and holes are then drifting towards the detector electrodes, then leaving the detector. During this drift, the electrons and holes induce an electrical current in the electrode, a current which may be measured, e.g. through a Charge Sensitive Amplifier (CSA), followed by a Shaping Filter (SF).
  • CSA Charge Sensitive Amplifier
  • SF Shaping Filter
  • the total charge in one induced current pulse is proportional to this energy.
  • the current pulse is amplified in the CSA and then filtered by the SF filter.
  • the pulse amplitude after filtering is proportional to the total charge in the current pulse, and therefore proportional to the x-ray energy.
  • the pulse amplitude is measured by comparing its value with one or several threshold values (Thr) in one or more comparators (COMP), and counters are introduced by which the number of cases when a pulse is larger than the threshold value may be recorded. In this way it is possible to count and/or record the number of X-ray photons with an energy exceeding an energy corresponding to respective threshold value (Thr) which has been detected within a certain time frame.
  • a so-called energy-discriminating detector is obtained, in which the detected photons can be sorted into energy bins corresponding to the various threshold values.
  • this type of detector is also referred to as a multi-bin detector.
  • the energy information allows for new kinds of images to be created, where new information is available and image artifacts inherent to conventional technology can be removed.
  • the pulse heights are compared to a number of programmable thresholds in the comparators and classified according to pulse-height, which in turn is proportional to energy.
  • a first, simplified functional for example a penalized weighted least squares functional
  • a second, more complicated functional may be a good approximation to the optimum of a second, more complicated functional.
  • a projection-based functional is able to model the image acquisition more accurately than an image-based functional. Therefore, it is preferable to let the first, simplified functional, be a penalized weighted least squares functional.
  • the publication " Multi-Material Decomposition Using Statistical Image Reconstruction for Spectral CT" by Y. Long and J. Fessler, IEEE Trans. Med. Imag. 33, pp. 1614-1626 (2014 ), relates to a method of using a set of basis images resulting from image-based material decomposition to initialize an optimization of a functional which contains a more complete model of the imaging system, including a model of Poisson noise.
  • the method of the publication does not include using basis images resulting from the optimization of a projection-based functional to initialize the optimization.
  • a method of reconstructing image data from x-ray data measured with an imaging system comprising at least one photon-counting detector.
  • the method comprises the step of obtaining S1 a representation of data measured by the photon-counting detector.
  • the method also comprises the step of generating S2 first image data based on a projection based first functional using a first algorithm, the projection based first functional being dependent on the representation of data.
  • the method further comprises the step S3 of updating, based on a second functional that includes a model of at least one physical effect not included in the projection based first functional, the first image data to obtain second image data.
  • the method is schematically illustrated in the flow diagram of FIG.3 .
  • the proposed method may for example be based on first solving a simpler problem, e.g., the problem of solving the functional defined by formula (3) above, by using a first optimization algorithm and then applying at least one update so that the solution becomes a better approximation of the full MAP problem defined by formula (2) above.
  • it builds on obtaining first image data by optimizing a first, simplified functional and subsequently applying one or more update steps so that the resulting image data is a better approximation than the first image data to the optimum of a second functional, which models additional physical effects not modelled by the first functional.
  • the proposed technology provides a method for reconstructing images based on measurements performed by a photon counting detector.
  • the measurements are provided, in a step S1 as input in order to be able to generate, in a step S2, image data based on a first projection based functional.
  • the measurement data is provided in a representation that is suitable for computing the first functional.
  • the measurement data may thus be provided in a pre-processed form, i.e. the raw measurement data may have been subject to pre-processing or pre-handling whereby a particular measurement data representation is obtained.
  • the particular representation should be chosen so as to obtain a form that is suitable for use as input in the first functional in order to generate the first image data.
  • the measurement data may in particular be tomographic x-ray data measured with an imaging system comprising at least one photon-counting detector.
  • the method also comprises to update the generated first image data in order to obtain second image data.
  • the update may be done in several ways which will be described below.
  • the updates are however all done based on a second functional that provides a more detailed model of the physical effects present during x-ray detection.
  • a particular example illustrating the differences between the first functional and a second, more detailed, functional is given by the functional defined by formula (3) and the functional defined by formula (2) provided earlier.
  • the step S2 of generating first image data might comprise using a first algorithm on the first functional where the first functional is a penalized weighted least squares functional.
  • the step S2 of generating first image data comprises using a first algorithm on the first functional where the first functional is a maximum likelihood or a maximum a posteriori functional.
  • Still another embodiment provides a method wherein the first algorithm, also referred to as a first optimization algorithm, is a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • the first algorithm also referred to as a first optimization algorithm
  • the first algorithm is a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • the optimum of the simplified functional which is easy to compute, can be corrected in one or a small number of update steps, to yield an approximate optimum of the second functional.
  • a large number of iterations of an optimization algorithm converging to the optimum of the second functional which could be computationally expensive, is avoided.
  • the invention provides a method where the update is performed in a first manner utilizing a particular term from the second functional.
  • the invention provides a method wherein the step S3 of updating the first image data to obtain second image data comprises to perform an image updating algorithm on the first functional when at least one term from the second functional has been added to the first functional, the at least one term providing a model of at least one physical effect not included in the first functional.
  • the first functional may be used in its original form, see e.g. the functional representation provided by formula (3) above, to yield first image data a.
  • This first image data a may then be updated by adding a particular term from the second functional, see e.g. functional representation provided by formula (3) for a particular second functional, that provides a model of a physical effect not included in the first functional.
  • This particular term may be seen as a perturbation of the first functional, i.e. the first functional is perturbed by the particular term from the second functional.
  • it will be possible to update the first image data a by inserting it into the perturbed first functional and perform an image updating algorithm in order to yield second image data a*.
  • the physical effects modelled by the particular term may include a model of the Poisson noise, pile-up, optical blur due to the finite size of the detector and focal spot, or detector cross-talk. It is possible to add several distinct terms from the second functional to the first functional. The different terms may model several different physical effects.
  • the image updating algorithm used for the initial first functional e.g. the non-perturbed first functional, may be a different image updating algorithm to the image updating algorithm that is used to obtain updated image data by applying it to the perturbed first functional. This embodiment provides a way for updating the image data by using a particular term, or particular terms, from the second functional.
  • the disclosed method works best if the second functional can be regarded as a small perturbation of the simplified functional. This is often the case if the second functional, for example, is based on a representation of one or several physical effects that are not modelled by the simplified functional.
  • the physical effects may include a model of the Poisson noise, pile-up, optical blur due to the finite size of the detector and focal spot, or detector cross-talk.
  • the step of updating the first image data comprises to perform a series expansion of at the optimum of the perturbed first functional around the first image data with respect to at least one parameter describing the magnitude of at least one particular term in the second functional and compute, based on the series expansion, updated image data where at least one term in the series expansion of the optimum has been added to the first image data to thereby obtain second image data.
  • the update comprises the addition of at least one term in a series expansion that converges to an optimum of the second functional.
  • an optimum of a functional refers to either a maximum point or a minimum point.
  • a maximum point of a functional L is an input data vector a of the functional L such that the value L( a ) of the functional evaluated with the data vector as input is larger than the value L( a' ) of the functional evaluated for any other input data vector a' in a set of permissible input data vectors.
  • a minimum point of a functional L is an input data vector a of the functional L such that the value L (a) of the functional evaluated with the data vector as input is smaller than the value L (a') of the functional evaluated for any other input data vector a' in a set of permissible input data vectors.
  • ⁇ ( a , m ) denotes the target functional to be minimized in (2) and ⁇ 0 ( a , m ) the target functional in (3)
  • ⁇ ( a , m ) can be expressed as ⁇ 0 ( a , m ) + d ⁇ 1 ( a , m ) where d is a perturbation strength parameter.
  • a h 0 m
  • H ⁇ 0 is the Hessian matrix of ⁇ 0 with respect to a .
  • Solving this quadratic equation for h 1 typically requires an amount of computational power comparable to two iterations with an iterative method for solving (3). Since direct iterative optimization of (2) may require hundreds of iterations, this can be very time-saving. If necessary, higher order corrections can be calculated similarly.
  • perturbation strength parameter d in certain embodiments may be set to one after the Taylor expansion has been performed.
  • the computation of the first functional, when perturbed by terms from the second functional yields a converging solution.
  • the first few terms that are computed may yield a good approximation. In the latter cases it may not be necessary to set d to one.
  • the proposed method can combine the ability of the full MAP method (2) to model Poisson noise, detector cross-talk and object scatter with the speed of the penalized weighted least squares problem (3).
  • the fact that a well-established reconstruction method may be used as a part of the proposed method also facilitates its introduction in clinical CT scanners.
  • the second algorithm referred to above may optionally be a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • the image data can be updated in several steps, such that the result of a first set of updates with a first update strategy is used as input to a second set of updates with a second update strategy.
  • the above proposed algorithms may thereby be combined sequentially.
  • a method that can be used to select particular image data that can be used as reconstructed image data. To this end there is provided a method that also comprises the steps of:
  • One purpose of this particular embodiment is to select the particular image data that best approximates the optimum of the second functional.
  • a representation of the measured data is obtained in a step S1.
  • first image data is generated based on a projection based first functional, denoted L1 using a first algorithm and the obtained representation of data.
  • first data referred to as a
  • L2 the second functional
  • first functional L1 does not contain the terms used as perturbations for the first functional, i.e. the perturbation terms are comprised in ⁇ L.
  • ⁇ L includes terms that model of at least one physical effect that is not included in the projection based first functional.
  • One particular way is to add one or several terms comprised in ⁇ L to the first functional in order to get L1+ ⁇ L.
  • one possibility is to expand one or several of the terms comprised in ⁇ L in a Taylor series and add at least one of the terms to L1.
  • This perturbed first functional may now be subject to an algorithm, either different from the one used to generate first image data or the same algorithm, whereby second image data, referred to as a*, is obtained by updating the first image data in the step S3.
  • This second algorithm may exemplarily be an iterative algorithm.
  • the second algorithm may include Taylor expanding the optimum of the perturbed first functional, viewed as a function of the input data, around the first image data a. The Taylor expansion is thereby made with respect to a parameter d determining the magnitude of the perturbation term. From this Taylor expansion, one or several correction term can be computed, which are applied to a and thereby yield a*.
  • second image data will be obtained by applying an algorithm to the perturbed first functional.
  • first and second image data a and a* .
  • the method now proceeds with the aim of determining which particular image data of the first and second image data that best approximates the second functional. That is, the method proceeds and determines which of the first and second image data that best approximates the complete solution to the second functional. This may also be seen as determining which of the first and second image data that converges to a solution to the second functional.
  • This may be obtained by first computing, in a step S4, the value of the second functional L2 when the second functional depends on the first image data, i.e.
  • L2(a) and when it depends on the second image data i.e. L2 (a*).
  • the computed values L2(a) and L2 (a*) are compared in a step S5 in order to select, in a step S6, image data to be used as reconstructed image data based on the comparison.
  • a functional that is evaluated based on a particular image data which might be a vector or even a vector valued function, provides a scalar as output.
  • the purpose of the comparing step and selecting step is to select the particular image data that best approximates a solution to the second functional.
  • a first image data vector a 1 is a better approximation than a second image data vector a 2 to a third image data vector a 3 if a mathematical measure of the discrepancy between a 1 and a 2 is less than the mathematical measure of the discrepancy between a 2 and a 3 .
  • a mathematical measure of the discrepancy between a and a' may for example be an Euclidian norm of a - a' , or a weighted Euclidean norm of a - a' , or an L p norm of a - a the absolute value of the largest element of a - a' , or the Kullback-Leibler divergence between a - a' .
  • the selected image data may be used as input in an updating scheme for updating the reconstructed image data.
  • the image data can be updated in several steps, such that the result of a first set of updates with a first update strategy is used as input to a second set of updates with a second update strategy.
  • the above proposed algorithms may thereby be combined sequentially.
  • An optional embodiment of the methods described above provides a method wherein the step of computing the value of the second functional comprises to compute an estimate of the value second functional by performing at least one step in an algorithm.
  • the first image data and the second image data may be fed into the second functional and be subjected to one or several steps in an algorithm in order to yield an approximation of the value of the second functional.
  • the particular image data to be used as reconstructed image data may then be selected, in step S6, based on the obtained approximation.
  • a particular version of the proposed embodiment provides a wherein the step S6 of selecting the image data to be used as input comprises to select the image data that yields the lowest or highest value for the second functional.
  • the value of the second functional is also intended an approximate value of the second functional that may have been obtained by performing one or several steps of an algorithm applied to the second functional.
  • the at least one update of the first image data may be one or several iteration steps of a second optimization algorithm, such that the output of the algorithm converges to the optimum of the second functional.
  • the steps of the second optimization algorithm may be more computationally heavy than the first algorithm, so that it is preferable to use the first algorithm to optimize the first functional and one or a few steps of the second algorithm to update the image data, thereby approximating the optimum of the second functional.
  • the second optimization algorithm may be a Newton method, which is may be computationally expensive but still possible to use for a small number of iterations.
  • the at least one update comprises the first part of a systematic update scheme the result of which, if the updates are applied repeatedly, converges to the solution of the second functional.
  • a systematic update scheme is meant a set of rules which allows calculation of a series of updated sets of image data, from a first set of image data and, possibly, also from the result of applying previous updates to a first set of image data.
  • Such an update scheme is different from ad-hoc corrections since it will converge to the statistically optimal image estimator provided that the second functional, for example the full problem (2) can be regarded as a small perturbation of the first functional, for example (3).
  • the physical effect modelled by the second functional may in all of the described embodiments relate to one of the following, or a combination thereof: Poisson noise statistics, optical blur, pile-up, detector element cross-talk and object scatter.
  • the first image data comprises at least one basis material image.
  • This particular embodiment utilizes a technique enabled by energy-resolved x-ray imaging. That is, a technique commonly referred to as basis material decomposition.
  • Basis material decomposition has been described in Alvarez and Macovski, "Energy-selective reconstructions in X-ray computerised tomography", Phys. Med. Biol. 21, 733.
  • the line integral A i of each of the basis coefficients a i is inferred from the measured data in each projection ray l from the source to a detector element.
  • ⁇ i is the expected number of counts in energy bin i
  • E is the energy
  • S i is a response function which depends on the spectrum shape incident on the imaged object, the quantum efficiency of the detector and the sensitivity of energy bin i to x-rays with energy E .
  • energy bin is most commonly used for photon-counting detectors, this formula can also describe other energy resolving x-ray systems such as multi-layer detectors or kVp switching sources.
  • the maximum likelihood method may be used to estimate A i under the assumption that the number of counts in each bin is a Poisson distributed random variable. This is accomplished by minimizing the negative log-likelihood function, see Roessl and Proksa, K-edge imaging in x-ray computed tomography using multi-bin photon counting detectors, Phys. Med. Biol.
  • a tomographic reconstruction to obtain the basis coefficients a i may be performed.
  • This procedural step may be regarded as a separate tomographic reconstruction, or may alternatively be seen as part of the overall basis decomposition.
  • the result is a material specific projection image, also called a basis image, for each basis i .
  • This basis image can either be viewed directly (e.g. in projection x-ray imaging) or taken as input to a reconstruction algorithm to form maps of basis coefficients a i inside the object (e.g. in CT).
  • the result of a basis decomposition can be regarded as one or more basis image representations, such as the basis coefficient line integrals or the basis coefficients themselves.
  • the first image data may be obtained by applying a first optimization algorithm on the first functional.
  • the first optimization algorithm is a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • the update of the first image data to obtain second image data may in the same example be obtained by the application of at least one iteration of a second iterative method optimizing the second functional.
  • the second iterative method is a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • the update of the first image data may moreover comprise the addition of at least one term in a series expansion converging to an optimum of the second functional.
  • the second optimization method may be an iterative coordinate descent method, a separable quadratic surrogates method (also called a separable paraboloidal surrogates method), an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of these.
  • the full Poisson noise model can be included in the optimization problem that is solved iteratively, using for example a pixel-wise separable quadratic surrogates (PWSQS) method similar to that described in Y. Long and J. Fessler, "Multi-Material Decomposition Using Statistical Image Reconstruction for Spectral CT", IEEE Trans. Med. Imag. 33, pp. 1614-1626 (2014 ).
  • PWSQS pixel-wise separable quadratic surrogates
  • the detector cross-talk alters the image so much that a few terms in the series expansion are insufficient to correct for it.
  • the detector cross-talk must be included in the optimization of the first functional by including a blur operator in (3), while the series expansion is used for taking the Poisson noise model and object scatter into account.
  • the initial optimization may for example be made using an optimization algorithm based on the separable quadratic surrogates method.
  • both Poisson noise and detector cross-talk may be represented in the optimization of the first functional, whereas object scatter, which is too expensive to compute in each iteration, can be corrected for using the series expansion method.
  • FIG.4 where a flowchart of an exemplary implementation of an iterative reconstruction algorithm is shown.
  • the measured data or a processed version of the measured data, is compared with the forward projection of the current image estimate.
  • a back-projection step is used together with prior information about the expected content of the image to calculate an image update which is applied to the current image estimate to form a new image estimate.
  • This procedure is iterated until a stopping criterion is fulfilled.
  • Such an iterative reconstruction algorithm may exemplarily be used in the present invention for generating S2 first image data or for updating S3 the first image data to obtain second image data.
  • Measured data is obtained from a photon-counting detector and processed to yield preprocessed data.
  • Said preprocessed data is then used as input to a first reconstruction algorithm which is based on a first functional and generates a first reconstructed image as output.
  • the first reconstructed image is then updated in an image update step, where the update is based on a second functional, thereby yielding a final reconstructed image.
  • Measured data is obtained from a photon-counting detector and processed to yield preprocessed data.
  • the preprocessed data is then used as input to a first reconstruction algorithm which optimizes a first functional using an iterative algorithm and generates first reconstructed image data as output.
  • a second functional which includes a model of one or more physical effects not included in the first functional.
  • image corrections are calculated. The image corrections are then applied to the first reconstructed image data to yield final reconstructed image data.
  • FIG.1 where it is illustrated how an x-ray imaging system 100 comprising an image processing device 30 connected to an x-ray system 20 adapted for detecting x-rays transmitted from an x-ray source 20 .
  • the x-ray system provides the image processing device 30 with representations of the measured data.
  • the representations may be any representation that is suitable as input in an image reconstruction process. The particular ways and representations are well known in the art.
  • an image processing device 30 for reconstructing image data from x-ray data measured with an imaging system comprising at least one photon-counting detector.
  • the image processing device 30 is configured to obtain a representation of data measured by the photon-counting detector.
  • the image processing device 30 is also configured to generate first image data based on a projection based first functional using a first algorithm, the projection based first functional being dependent on the representation of data.
  • the image processing device 30 is also configured to update, based on a second functional that includes a model of at least one physical effect not included in the projection based first functional, the first image data to obtain second image data.
  • the image processing device 30 is further configured to update the first image data by being configured to perform a series expansion of the optimum of the perturbed first functional as a function of the input data, with respect to at least one parameter determining the magnitude of at least one particular term in the second functional which is not present in the first functional, and configured to compute a correction to the first image data based on the series expansion, to thereby obtain second image data.
  • an image processing device configured to generate first image data by using a first algorithm on the first functional where the first functional is a penalized weighted least squares functional.
  • an image processing device configured to generate first image data by using a first algorithm on the first functional where the first functional is a maximum likelihood or a maximum a posteriori functional.
  • the first algorithm is a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • an image processing device configured to update the first image data to obtain second image data by performing an image updating algorithm on the first functional when at least one term from the second functional has been added to the first functional, thereby yielding a perturbed first functional, the at least one term providing a model of at least one physical effect not included in the first functional.
  • a possible embodiment of the proposed image processing device provides an image processing device wherein the second algorithm is a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • the second algorithm is a Newton method, an iterative coordinate descent method, a separable quadratic surrogates method, an expectation maximization method, a conjugate gradient method, an ordered subset method, or a combination of a plurality of the methods.
  • the image processing device is also configured to compare the computed values and select, based on the comparison, image data to be used as reconstructed image data.
  • Yet another embodiment of the proposed technology provides an image processing device that is configured to use the selected image data as input in an updating scheme for updating the reconstructed image data.
  • an image processing device that is configured to compute the value of the second functional by being configured to compute an estimate of the value of the second functional by performing at least one step in an algorithm.
  • the proposed technology provides an image processing device that is configured to select the image data to be used as input by selecting the image data that yields the lowest or the highest value for the second functional.
  • a particular embodiment of the proposed technology provides an image processing device wherein the physical effect is one of the following, or a combination thereof: Poisson noise statistics, optical blur, pile-up, detector element cross-talk and object scatter.
  • a preferred embodiment of the proposed technology provides an image processing device wherein the first image data comprises at least one basis material image.
  • the image processing device comprises at least one processor 110 and memory 120 , the memory 120 comprising instructions, which when executed by the at least one processor 110 , cause the at least one processor 110 to reconstruct image data from x-ray data measured with an imaging system comprising at least one photon-counting detector.
  • the arrangement 100 may also include a communication circuit 130 .
  • the communication circuit 130 may include functions for wired and/or wireless communication with other devices and/or network nodes in the network.
  • the communication circuit 130 may be based on radio circuitry for communication with one or more other nodes, including transmitting and/or receiving information.
  • the communication circuit 130 may be interconnected to the processor 110 and/or memory 120 .
  • the communication circuit 130 may include any of the following: a receiver, a transmitter, a transceiver, input/output (I/O) circuitry, input port(s) and/or output port(s).
  • FIG.8 also provides a schematic illustration of an image processing device 30 that comprises a communication circuitry 130 .
  • the arrangement may be based on a hardware circuitry implementation.
  • suitable hardware circuitry include one or more suitably configured or possibly reconfigurable electronic circuitry, e.g. Application Specific Integrated Circuits (ASICs), Field Programmable Gate Arrays (FPGAs), or any other hardware logic such as circuits based on discrete logic gates and/or flip-flops interconnected to perform specialized functions in connection with suitable registers (REG) and/or memory units (MEM).
  • ASICs Application Specific Integrated Circuits
  • FPGAs Field Programmable Gate Arrays
  • REG registers
  • MCM memory units
  • FIG.7 is a schematic diagram illustrating an example of a computer-implementation 200 according to an embodiment.
  • a computer program 225; 235 which is loaded into the memory 220 for execution by processing circuitry including one or more processors 210 .
  • the processor(s) 210 and memory 220 are interconnected to each other to enable normal software execution.
  • An optional input/output device 240 may also be interconnected to the processor(s) 210 and/or the memory 220 to enable input and/or output of relevant data such as input parameter(s) and/or resulting output parameter(s).
  • processor' should be interpreted in a general sense as any system or device capable of executing program code or computer program instructions to perform a particular processing, determining or computing task.
  • the processing circuitry including one or more processors 210 is thus configured to perform, when executing the computer program 225 , well-defined processing tasks such as those described herein.
  • the processing circuitry does not have to be dedicated to only execute the above-described steps, functions, procedure and/or blocks, but may also execute other tasks.
  • the computer program is adapted to reconstruct image data from x-ray data measured with an imaging system comprising at least one photon-counting detector and comprises instructions, which when executed by at least one processor, cause the processor(s) to:
  • the proposed technology also provides a carrier comprising the computer program, wherein the carrier is one of an electronic signal, an optical signal, an electromagnetic signal, a magnetic signal, an electric signal, a radio signal, a microwave signal, or a computer-readable storage medium.
  • a computer-program product comprising a computer-readable medium 220; 230 having stored thereon a computer program 225; 235 .

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Claims (14)

  1. Verfahren zum Rekonstruieren von Bilddaten aus Röntgendaten, die mit einem Bildgebungssystem gemessen werden, das mindestens einen photonenzählenden Detektor umfasst, wobei das Verfahren folgende Schritte umfasst:
    - Erzielen (S1) einer Darstellung von Daten, die von dem photonenzählenden Detektor gemessen werden;
    - Generieren (S2) von ersten Bilddaten basierend auf einem projektionsbasierten ersten Funktional unter Verwendung eines ersten Algorithmus, wobei das projektionsbasierte erste Funktional von der Darstellung der Daten abhängig ist;
    - Aktualisieren (S3) der ersten Bilddaten basierend auf einem zweiten Funktional, das ein Modell mindestens eines physikalischen Effekts umfasst, der nicht in dem projektionsbasierten ersten Funktional enthalten ist, um zweite Bilddaten zu erzielen,
    wobei der Schritt (S3) des Aktualisierens der ersten Bilddaten das Ausführen einer Reihenentwicklung des Bestwertes des ersten Funktionals, wenn mindestens ein Term aus dem zweiten Funktional zu dem ersten Funktional addiert wurde, wobei die Reihenentwicklung mit Bezug auf mindestens einen Parameter erfolgt, der die Größe mindestens eines bestimmten Terms aus dem zweiten Funktional bestimmt, und das Berechnen basierend auf der Reihenentwicklung einer Aktualisierung der ersten Bilddaten, um dadurch zweite Bilddaten zu erzielen, umfasst.
  2. Verfahren nach Anspruch 1, wobei der Schritt (S2) des Generierens von ersten Bilddaten das Verwenden eines ersten Algorithmus an dem ersten Funktional umfasst, wobei das erste Funktional ein penalisiertes gewichtetes Funktional der kleinsten Quadrate, ein Funktional der höchsten Wahrscheinlichkeit oder ein maximales A-posteriori-Funktional ist.
  3. Verfahren nach einem der Ansprüche 1 bis 2, wobei der Schritt (S3) des Aktualisierens der ersten Bilddaten, um zweite Bilddaten zu erzielen, das Ausführen eines Bildaktualisierungsalgorithmus an dem ersten Funktional umfasst, wenn mindestens ein Term aus dem zweiten Funktional zu dem ersten Funktional addiert wurde, wobei der mindestens eine Term ein Modell mindestens eines physikalischen Effekts, der nicht in dem ersten Funktional enthalten ist, bereitstellt.
  4. Verfahren nach einem der Ansprüche 1 bis 3, wobei das Verfahren ferner folgende Schritte umfasst:
    - Berechnen (S4) des Wertes des zweiten Funktionals, wenn das zweite Funktional:
    a) von den ersten Bilddaten abhängig ist, und
    b) von den zweiten Bilddaten abhängig ist; und
    - Vergleichen (S5) der berechneten Werte, und
    - Auswählen (S6), basierend auf dem Vergleich, von Bilddaten, die als rekonstruierte Bilddaten zu verwenden sind.
  5. Verfahren nach Anspruch 4, wobei die ausgewählten Bilddaten als Eingabe in ein Aktualisierungsschema zum Aktualisieren der rekonstruierten Bilddaten verwendet werden.
  6. Verfahren nach einem der Ansprüche 1 bis 5, wobei die ersten Bilddaten mindestens ein Basismaterialbild umfassen.
  7. Bildverarbeitungsvorrichtung zum Rekonstruieren von Bilddaten aus Röntgendaten, die mit einem Bildgebungssystem gemessen werden, das mindestens einen photonenzählenden Detektor umfasst, wobei:
    - die Bildverarbeitungsvorrichtung dazu konfiguriert ist, eine Darstellung von Daten zu erzielen, die von dem photonenzählenden Detektor gemessen werden;
    - die Bildverarbeitungsvorrichtung dazu konfiguriert ist, erste Bilddaten basierend auf einem projektionsbasierten ersten Funktional unter Verwendung eines ersten Algorithmus zu generieren, wobei das projektionsbasierte erste Funktional von der Datendarstellung abhängig ist;
    - die Bildverarbeitungsvorrichtung dazu konfiguriert ist, basierend auf einem zweiten Funktional, das ein Modell mindestens eines physikalischen Effekts, der nicht in dem projektionsbasierten ersten Funktional enthalten ist, umfasst, die ersten Bilddaten zu aktualisieren, um zweite Bilddaten zu erzielen,
    wobei die Bildverarbeitungsvorrichtung dazu konfiguriert ist, die ersten Bilddaten durch Ausführen einer Reihenentwicklung des Bestwertes des ersten Funktionals zu aktualisieren, wenn mindestens ein Term des zweiten Funktionals zu dem ersten Funktional addiert wurde, wobei die Reihenentwicklung mit Bezug auf mindestens einen Parameter erfolgt, der die Größe mindestens eines bestimmten Terms aus dem zweiten Funktional bestimmt, und dazu konfiguriert ist, basierend auf der Reihenentwicklung eine Aktualisierung der ersten Bilddaten zu berechnen, um dadurch zweite Bilddaten zu erzielen.
  8. Bildverarbeitungsvorrichtung nach Anspruch 7, wobei die Bildverarbeitungsvorrichtung dazu konfiguriert ist, erste Bilddaten unter Verwendung eines ersten Algorithmus an dem ersten Funktional zu generieren, wobei das erste Funktional ein penalisiertes gewichtetes Funktional der kleinsten Quadrate, ein Funktional der höchsten Wahrscheinlichkeit oder ein maximales A-posteriori-Funktional ist.
  9. Bildverarbeitungsvorrichtung nach einem der Ansprüche 7 bis 8, wobei die Bildverarbeitungsvorrichtung dazu konfiguriert ist, die ersten Bilddaten zu aktualisieren, um zweite Bilddaten zu erzielen durch Ausführen eines Bildaktualisierungsalgorithmus an dem ersten Funktional, wenn mindestens ein Term aus dem zweiten Funktional zu dem ersten Funktional addiert wurde, wobei der mindestens eine Term ein Modell mindestens eines physikalischen Effekts, der nicht in dem ersten Funktional enthalten ist, bereitstellt.
  10. Bildverarbeitungsvorrichtung nach einem der Ansprüche 7 bis 9, wobei:
    - die Bildverarbeitungsvorrichtung dazu konfiguriert ist, den Wert des zweiten Funktionals zu berechnen, wenn das zweite Funktional:
    a) von den ersten Bilddaten abhängig ist, und
    b) von den zweiten Bilddaten abhängig ist; und
    - die Bildverarbeitungsvorrichtung dazu konfiguriert ist, die berechneten Werte zu vergleichen, und
    - die Bildverarbeitungsvorrichtung dazu konfiguriert ist, basierend auf dem Vergleich, Bilddaten auszuwählen, die als rekonstruierte Bilddaten zu verwenden sind.
  11. Bildverarbeitungsvorrichtung nach Anspruch 10, wobei die Bildverarbeitungsvorrichtung dazu konfiguriert ist, die ausgewählten Bilddaten als Eingabe in ein Aktualisierungsschema zum Aktualisieren der rekonstruierten Bilddaten zu verwenden.
  12. Bildverarbeitungsvorrichtung nach einem der Ansprüche 7 bis 11, wobei die ersten Bilddaten mindestens ein Basismaterialbild umfassen.
  13. Bildverarbeitungsvorrichtung nach einem der Ansprüche 7 bis 12, wobei die Bildverarbeitungsvorrichtung mindestens einen Prozessor und einen Speicher umfasst, wobei der Speicher Anweisungen umfasst, die bewirken, wenn sie von dem mindestens einen Prozessor ausgeführt werden, dass der mindestens eine Prozessor Bilddaten aus Röntgendaten rekonstruiert, die mit einem Bildgebungssystem gemessen werden, das mindestens einen photonenzählenden Detektor umfasst.
  14. Computerprogramm, das dazu geeignet ist, Bilddaten aus Röntgendaten zu rekonstruieren, die mit einem Bildgebungssystem gemessen werden, das mindestens einen photonenzählenden Detektor umfasst, das Anweisungen umfasst, die bewirken, wenn sie von mindestens einem Prozessor ausgeführt werden, dass der mindestens eine Prozessor ein Verfahren nach einem der Ansprüche 1 bis 6 ausführt.
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